Freshman's dream

Freshman's dream
The freshman's dream is a name sometimes given to the erroneous equation , where is a real number (usually a positive integer greater than 1) and are non-zero real numbers. Beginning students commonly make this error in computing the power of a sum of real numbers, falsely assuming powers distribute over sums. When n = 2, it is easy to see why this is incorrect: (x + y)2 can be correctly computed as x2 + 2xy + y2 using distributivity (commonly known by students in the USA as the FOIL method). For larger positive integer values of n, the correct result is given by the binomial theorem.
The name "freshman's dream" also sometimes refers to the theorem that says that for a prime number p, if x and y are members of a commutative ring of characteristic p, then
(x + y)p = xp + yp. In this more exotic type of arithmetic, the "mistake" actually gives the correct result, since p divides all the binomial coefficients apart from the first and the last, making all the intermediate terms equal to zero.
The identity is also actually true in the context of tropical geometry, where multiplication is replaced with addition, and addition is replaced with minimum.
Examples
, but .
does not equal . For example which does not equal . In this example, the error is being committed with the exponent .
Prime characteristic
When is a prime number and and are members of a commutative ring of characteristic , then . This can be seen by examining the prime factors of the binomial coefficients: the nth binomial coefficient is
The numerator is p factorial(!), which is divisible by p. However, when , both n! and are coprime with p since all the factors are less than p and p is prime. Since a binomial coefficient is always an integer, the nth binomial coefficient is divisible by p and hence equal to 0 in the ring. We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation.
Thus in characteristic p the freshman's dream is a valid identity. This result demonstrates that exponentiation by p produces an endomorphism, known as the Frobenius endomorphism of the ring.
The demand that the characteristic p be a prime number is central to the truth of the freshman's dream. A related theorem states that if p is prime then in the polynomial ring . This theorem is a key fact in modern primality testing.
History and alternate names
The history of the term "freshman's dream" is somewhat unclear. In a 1940 article on modular fields, Saunders Mac Lane quotes Stephen Kleene's remark that a knowledge of in a field of characteristic 2 would corrupt freshman students of algebra. This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic. Since then, authors of undergraduate algebra texts took note of the common error. The first actual attestation of the phrase "freshman's dream" seems to be in Hungerford's graduate algebra textbook (1974), where he quotes McBrien. Alternative terms include "freshman exponentiation", used in Fraleigh (1998). The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the 19th century.
Since the expansion of is correctly given by the binomial theorem, the freshman's dream is also known as the "child's binomial theorem" or "schoolboy binomial theorem".
See also
Pons asinorum
Primality test
Sophomore's dream
Frobenius endomorphism
References
Algebra education
Mathematical fallacies
Theorems in ring theory
Prime numbers